On the counting of k-digit terms in increasing sequences of positive integers
In this article, we prove that the sequence \(\left[b_k\right]_{k\geq 1}\) of the number \(b_k\) of k-digit terms in an increasing sequence of positive integers \(\left[a_n\right]_{n\geq 1}\) in a given base \(B\), satisfies the following limit of the ratios \(\lim_{k \to \infty} \frac{b_k}{b_{k-1}} = B^\alpha\), provided that the counting function of \(\left[a_n\right]_{n\geq 1}\) is regularly varying with index \(\alpha\). As corollaries, we obtain the cases of sequences of positive asymptotic density (\(\alpha=1\)) [1], of sparse sequences such as the primes (\(\alpha=1\) with a slowly varying factor \(\log\left(x\right)\)), and of polynomial sequences such as the squares or higher powers (\(\alpha=1/q\)).
Theorem. Let \(\left[a_n\right]_{n\geq 1}\) be an increasing sequence of positive integers in a given base \(B\ge 2\),
\(A(x) := \#\{n : a_n \le x\}\)
be its counting function, and
\(b_k := \#\{n : B^{k-1} \le a_n < B^k\}\)
the number of \(k\)-digit terms in \(\left[a_n\right]_{n\geq 1}\). If \(A\) is a regularly varying function with index \(\alpha>0\), i.e., \(A(x) \sim x^\alpha L(x) \; \text{as } x \to \infty\), for some slowly varying function \(L(x)\) (see [2]), then
\(\lim_{k\to\infty} \frac{b_k}{b_{k-1}} = B^\alpha\).
Proof. Since \(A\) is regularly varying function with index \(\alpha\), we have
\(A(B^k) \sim B^{k\alpha} L(B^k)\)
and
\(A(B^{k-1}) \sim B^{(k-1)\alpha} L(B^{k-1})\).
By the definition of \(b_k\) and the property that for every fixed \(\lambda >0\), \(L(\lambda x)=L(x)\), we obtain
\(b_k = A(B^k-1) – A(B^{k-1}) \sim A(B^k) – A(B^{k-1})\)
\(\sim B^{k\alpha} L(B^k) – B^{(k-1)\alpha}L(B^{k-1}) = B^{(k-1)\alpha} (B^\alpha L(B^k) – L(B^{k-1}))\)
\(\sim B^{(k-1)\alpha} (B^\alpha – 1) L(B^k)\)
and similarly
\(b_{k-1} \sim B^{(k-2)\alpha} (B^\alpha – 1) L(B^{k-1})\).
Using the property of a slowly varying function, \(L(B^k)/L(B^{k-1}) \to 1\) as \(k \to \infty\), we obtain
\(\lim_{k\to\infty} \frac{b_k}{b_{k-1}} = \lim_{k\to\infty} B^\alpha\frac{L(B^k)}{L(B^{k-1})} = B^\alpha\).
Q.E.D.
Corollary 1. Let \(\left[a_n\right]_{n\geq 1}\) be an increasing sequence of positive integers with positive asymptotic density, i.e.
\(\lim_{x \to \infty} \frac{A(x)}{x} = \delta > 0\).
Then \(A(x) \sim \delta x\) is regularly varying with index \(\alpha=1\) and slowly varying function \(L(x)=\delta\), and therefore
\(\lim_{k \to \infty} \frac{b_k}{b_{k-1}} = B\)
and
\(b_k \sim \delta\cdot \left(B-1\right)\cdot B^{k-1}\).
Corollary 2. Let \(\left[p_n\right]_{n\geq 1}\) be the sequence of prime numbers. By the Prime Number Theorem (see [3, 4]),
\(A(x) = \pi(x) \sim \frac{x}{\log x}\),
which is regularly varying with index \(\alpha=1\) and slowly varying function \(L(x) = \left(\log x\right)^{-1}\) (see [2]). Therefore,
\(\lim_{k \to \infty} \frac{b_k}{b_{k-1}} = B = 10\),
where \(b =\) A006879.
Remark. Theorem holds also in the case that \(b_k := \#\{n : a_n < B^k\}\). In such a case, Corollary 2 also holds for \(b =\) A006880. The proof is left as an exercise for the reader.
Corollary 3. Let \(\left[a_n\right]_{n\geq 1}\) be a polynomial sequence, e.g., \(a_n = n^q\) with \(q \ge 2\). Then
\(A(x) \sim x^{1/q}\),
which is regularly varying with index \(\alpha = 1/q\) (see [2]). Hence,
\(\lim_{k \to \infty} \frac{b_k}{b_{k-1}} = B^{1/p}\).
References
- OEIS Foundation Inc. (2025), Entries A337856, A346509, A346952, A347255, A347749, A348055, A348547, and A348549 in The On-Line Encyclopedia of Integer Sequences, http://oeis.org/
- N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press (1989).
- P. L. Chebyshev, Sur la totalité des nombres premiers inférieurs à une limite donnée, Journal de Mathématiques Pures et Appliquées, 17, 341-365 (1852).
- J. Elliott, Asymptotic Expansions of the Prime Counting Function, arXiv:1809.06633 [math.NT], 2024.
- OEIS Foundation Inc. (2025), Entries A006879 and A006880 in The On-Line Encyclopedia of Integer Sequences, http://oeis.org/
©Stefano Spezia. This work is licensed under a Creative Commons Attribution 4.0 International License
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